This 2nd edition textbook offers a rigorous introduction to measure theoretic probability with particular attention to topics of interest to mathematical statisticians-a textbook for courses in probability for students in mathematical statistics. It is recommended to anyone interested in the probability underlying modern statistics, providing a solid grounding in the probabilistic tools and techniques necessary to do theoretical research in statistics. For the teaching of probability theory to post graduate statistics students, this is one of the most attractive books available.

Of particular interest is a presentation of the major central limit theorems via Stein's method either prior to or alternative to a characteristic function presentation. Additionally, there is considerable emphasis placed on the quantile function as well as the distribution function. The bootstrap and trimming are both presented. Martingale coverage includes coverage of censored data martingales. The text includes measure theoretic preliminaries, from which the authors own course typically includes selected coverage.

This is a heavily reworked and considerably shortened version of the first edition of this textbook. "Extra" and background material has been either removed or moved to the appendices and important rearrangement of chapters has taken place to facilitate this book's intended use as a textbook.

New to this edition:

• Still up front and central in the book, Chapters 1-5 provide the "measure theory" necessary for the rest of the textbook and Chapters 6-7 adapt that measure-theoretic background to the special needs of probability theory
• Develops both mathematical tools and specialized probabilistic tools
• Chapters organized by number of lectures to cover requisite topics, optional lectures, and self-study
• Exercises interspersed within the text
• Guidance provided to instructors to help in choosing topics of emphasis

Autorentext

Galen Shorack, PhD, is Professor Emeritus in the Department of Statistics (of which he was a founding member) and Adjunct Professor in the Department of Mathematics at the University of Washington, Seattle. He received his Bachelor of Science and Master of Science degrees in Mathematics from the University of Oregon and his PhD in Statistics from Stanford University. Dr. Shorack's research interests include limit theorems in statistics, the theory of empirical processes, trimming-Winsorizing, and regular variation. He has served as Associate Editor of the Annals of Mathematical Statistics (Annals of Statistics) and is Fellow of the Institute of Mathematical Statistics.

Klappentext

The choice of examples used in this text clearly illustrate its use for a one-year graduate course. The material to be presented in the classroom constitutes a little more than half the text, while the rest of the text provides background, offers different routes that could be pursued in the classroom, as well as additional material that is appropriate for self-study. Of particular interest is a presentation of the major central limit theorems via Steins method either prior to or alternative to a characteristic function presentation. Additionally, there is considerable emphasis placed on the quantile function as well as the distribution function, with both the bootstrap and trimming presented. The section on martingales covers censored data martingales.

Inhalt

PrefaceUse of This TextDefinition of Symbols Chapter 1. Measures

1. Basic Properties of Measures

2. Construction and Extension of Measures

3. Lebesgue Stieltjes MeasuresChapter 2. Measurable Functions and Convergence

4. Mappings and s-Fields

5. Measurable Functions

6. Convergence

7. Probability, RVs, and Convergence in Law

8. Discussion of Sub s-FieldsChapter 3. Integration

9. The Lebesgue Integral

10. Fundamental Properties of Integrals

11. Evaluating and Differentiating Integrals

12. Inequalities

13. Modes of ConvergenceChapter 4 Derivatives via Signed Measures

14. Introduction

15. Decomposition of Signed Measures

16. The Radon Nikodym Theorem

17. Lebesgue's Theorem

18. The Fundamental Theorem of CalculusChapter 5. Measures and Processes on Products

19. Finite-Dimensional Product Spaces

20. Random Vectors on (O, ,P)

21. Countably Infinite Product Probability Spaces

22. Random Elements and Processes on (O, ,P)Chapter 6. Distribution and Quantile Functions

23. Character of Distribution Functions

24. Properties of Distribution Functions

25. The Quantile Transformation

26. Integration by Parts Applied to Moments

27. Important Statistical Quantities

28. Infinite VariancesChapter 7. Independence and Conditional Distributions

29. Independence

30. The Tail s-Field

31. Uncorrelated Random Variables

32. Basic Properties of Conditional Expectation

33. Regular Conditional ProbabilityChapter 8. WLLN, SLLN, LIL, and Series

34. Introduction

35. Borel Cantelli and Kronecker Lemmas

36. Truncation, WLLN, and Review of Inequalities

37. Maximal Inequalities and Symmetrization

38. The Classical Laws of Large Numbers (or, LLNs)

39. Applications of the Laws of Large Numbers

40. Law of the Iterated Logarithm (or, LIL)

41. Strong Markov Property for Sums of IID RVs

42. Convergence of Series of Independent RVs

43. Martinagles

44. Maximal Inequalities, Some with BoundariesChapter 9. Characteristic Functions and Determining Classes

45. Classical Convergence in Distribution

46. Determining Classes of Functions

47. Characteristic Functions, with Basic Results

48. Uniqueness and Inversion

49. The Continuity Theorem

50. Elementary Complex and Fourier Analysis

51. Esseen's Lemma

52. Distributions on Grids

53. Conditions for Ø to Be a Characteristic FunctionChapter 10. CLTs via Characteristic Functions

54. Introduction

55. Basic Limit Theorems

56. Variations on the Classical CLT

57. Examples of Limiting Distributions

58. Local Limit Theorems

59. Normality Via Winsorization and Truncation

60. Identically Distributed RVs

61. A Converse of the Classical CLT

62. Bootstrapping

63. Bootstrapping with Slowly WinsorizationChapter 11. Infinitely Divisible and Stable Distributions

64. Infinitely Divisible Distributions

65. Stable Distributions

66. Characterizing Stable Laws

67. The Domain of Attraction of a Stable Law

68. Gamma Approximations

69. Edgeworth ExpansionsChapter 12. Brownian Motion and Empirical Processes

70. Special Spaces

71. Existence of Processes on (C, C) and (D, D)

72. Brownian Motion and Brownian Bridge

73. Stopping Times

74. Strong Markov Property

75. Embedding a RV in Brownian Motion

76. Barrier Crossing Probabilities

77. Embedding the Partial Sum Process

78. Other Properties of Brownian Motion

79. Various Empirical Processes

80. Inequalities for the Various Empirical Processes

81. ApplicationsChapter 13. Martingales

82. Basic Technicalities for Martingales

83. Simple Optional Sampling Theorem

84. The Submartingale Convergence Theorem

85. Applications of the S-mg Convergence Theorem

86. Decomposition of a Submartingale Sequence

87. Optional Sampling

88. Applications of Optional Sampling

89. Introduction to Counting Process Martingales

90. CLTs for Dependent RVsChapter 14. Convergence in Law on Metric Spaces

91. Convergence in Distribution on Metric Spaces

92. Metrics for Convergence in Distribution Chapter 15. Asymptotics Via Empirical Processes

93. Introduction

94. Trimmed and Winsorized Means

95. Linear Rank Statistics and Finite Sampling

96. L-StatisticsAppendix A. Special DistributionsElementary ProbabilityDistribution Theory for StatisticsAppendix B. General Topology and Hilbert Space

97. General Topology

98. Metric Spaces

99. Hilbert SpaceAppendix C. More WLLN and CLT

100. Introduction

101. General Moment Estimation Specific

102. Slowly Varying Partial Variance when s2=8

103. Specific Tail Relationships

104. Regularly Varying Functions

105. Some Winsorized Variance Comparison

106. Inequalities for Winsorized Quantile…